Embodiments of the present invention relate to a method of processing data representing a wave propagating through a medium, in particular but not by way of limitation to a method of generating operators having a large timestep and embodying fine-scale medium properties. Embodiments of the present invention may be applied in the processing of seismic data, but the invention is not limited to this and embodiments of the present invention may also be applied to, for example, acoustic or electromagnetic imaging techniques, non-destructive testing, or medical imaging. Embodiments of the present invention also relates to a corresponding apparatus and computer-readable medium.
Modeling elastic wave propagation is the engine behind high-end seismic processing such as imaging and full-waveform inversion. In imaging, for example, seismic field data are back propagated into a model of the Earth by reversing the sense of time, as in reverse-time migration (RTM), and the method of this invention may be used for this purpose, especially where the Earth model contains complicated fine structure.
Prominent approaches to seismic modeling are the finite-difference (FD) method and the extended finite-element methods (spectral element (SEM) and discontinuous Gelerkin (DG)). While quite different in many ways, those approaches have in common the use of timesteps which are very small compared to the wave period. For example, in a finite difference method, the domain is partitioned in space to give a mesh or grid of points x0, x1 . . . xJ, separated by a spatial sampling interval Δx and is also partitioned in time to give a mesh or grid of time points t0, t1 . . . tN, separated by a time sampling interval or “timestep” Δt. A differential equation may then be converted to a series of difference equations that define a value of a quantity at a particular timepoint and a particular spatial point in terms of the values of that quantity at previous timepoints and/or previous spatial points. The value of the timestep is typically limited by a stability criterion and complicated spatial variations of the medium on a scale less than Δx are not admitted.